Why are RNNs better than MLPs at predicting time series data?
In an RNN, the output of the previous state is passed as an input to the current state. Intuitively, there is a temporal (time-based) relationship in the way in which input is processed in an RNN. It can understand how the current state was achieved on the basis of the previous values, i.e value at time-step $t$ is a result of value at time-steps $t-1, t-2$, and so on.
In a DNN, there is no temporal relationship in the way input is processed. Values at time-steps $t, t-1, t-2, \dots $ are all treated distinctly and not as a continuation of the previous time-step values.